the falsity of the Kalam Cosmological Argument

I have asserted that you cannot prove God, for to do so would deny faith. This does not mean, logically, that if you DID somehow prove God (for example, if God presented you with 10 Commandments, or a holy Qur'an) that God would vanish - that is not the point of the Douglas Adams excerpt either. The God that vanishes is the concept of God as an entity whose existence can be proved, rather than the requirement of God as an entity that must be believed in.

While technically true that God revealing Himself to man would constitute a proof of sorts, that also denies faith. Now the man to whom God has revealed Himself does not need faith. He can simply point to God. The believer however, will most likely never perceive God in that direct form, and thus their belief in God is a matter of faith and not proof.

I do believe that God exists, and that belief is independent of whether God proves his existence or not to me. Thus I do not need proof, and in fact God's direct revelation would not be proof. If I were to treat it as such, then I never truly had faith.

This is why an avowed atheist such as Steven Den Beste is my ally, against attempts to prove the existence of God, such as the Kalam Cosmological Argument. This supposed proof proceeds as such:

1. The universe either had (a) a beginning or (b) no beginning.


2. If it had a beginning, the beginning was either (a) caused or (b) uncaused.


3. If it had a cause, the cause was either (a) personal or (b) not personal.

The supposition is that if the Universe had a beginning, which was caused, and the cause was personal, then that proves God. Actually, it doesn't, because the only God the Kalam argument proves is the Deist God who created the Universe and then ceased all interference afterwards. That version of God is incompatible with Islam (or Judaism, or Christianity, according to my understanding).

But the methods by which the Kalam argument "proves" the answer to be (a) in each case constitute a gross abuse of mathematics and science. Note that by relying entirely on Math, Science, and Philosophy to attempt to infer the existence of an entity that transcends those concepts (by definition), the proof is already in violation of Godel's Incompleteness Theorem. But a more fundamental flaw is that the entire thesis rests on the postulate that "a trait of the actual infinite is that nothing can be added to it". They attempt to justify this using a thought-experiment called "Hilbert's Hotel" :

Let us imagine a hotel with a finite number of rooms, and let us assume that all the rooms are occupied. When a new guest arrives and requests a room, the proprietor apologises, 'Sorry--all the rooms are full.' Now let us imagine a hotel with an infinite number of rooms, and let us assume that again all the rooms are occupied. But this time, when a new guest arrives and asks for a room, the proprietor exclaims, 'But of course!' and shifts the person in room 1 to room 2, the person in room 2 to room 3, the person in room 3 to room 4, and so on... The new guest then moves into room 1, which has now become vacant as a result of these transpositions. But now let us suppose an infinite number of new guests arrive, asking for rooms. 'Certainly, certainly!' says the proprietor, and he proceeds to move the person in room 1 into room 2, the person in room 2 into room 4, the person in room 3 into room 6, the person in room 4 into 8, and so on... . In this way, all the odd-numbered rooms become free, and the infinity of new guests can easily be accommodated in them.

In this story the proprietor thinks that he can get away with his clever business move because he has forgotten that his hotel has an actually infinite number of rooms, and that all the rooms are occupied. The proprietor's action can only work if the hotel is a potential infinite, such that new rooms are created to absorb the influx of guests. For if the hotel has an actually infinite collection of determinate rooms and all the rooms are full, then there is no more room. (Craig, Kalam, 84-85)

The simplest counterexample is simply the infinite set N of real positive integers, ie { 1, 2, 3, ... }. Now all I do is add zero. We now have { 0, N} where N = { 1, 2, 3, ...}, ie {0, 1, 2, 3, ...}. There! We just added one to infinity.

To apply this counterexample against Hilbert's Hotel, simply consider the room numbers to be labeled according to N. Now the hypothetical guest arrives. Build him a new room with room number 0.

The distinction between a "potential" and "actual" infinite is arbitrary. The distinction is solely introduced to justify the claim - note the underlined sentence above in the Hilbert Hotel example, which explicitly tries to deflect my counterexample by labeling it a "potential" infinite. They are essentially trying to claim 1. The Hotel is a pure math construct, not in the Real World. 2. We cannot build a new room because the Hotel is in the Real World, not a pure math construct.

In fact, they repeat this type-confusion by asserting that because "potential infinities" cannot exist in the Real World, that this even applies to God. The implicit assumption that they (probably did not intend to) make is that God is bounded by the Real World. This may have been true for the Greek pantheon, but Allah is beyond human comprehension and the limits of his Creation.

UPDATE: Christopher Landsdown, a grad student in math, has some minor corrections. Most importantly, he demonstrates that - counter to assertions in teh comments below - an infinite subset of an infinite set is NOT equal to the infinite set. The Kalam people also really need to look up the difference between "isomorphism" and "homomorphism" in a group theory textbook.

The website invoking Kalam also argues that Cantor, of set-theory fame, also subscribed to a version of this argument and even tried to present it to the Pope:

In fact, until Gregor Cantor's work in set theory, mathematicians rejected the existence of an actual infinite as a mathematical concept. But Cantor himself denied the existential possibility of the actual infinite. In correspondence with the Pope, he even suggested that the existential impossibility of the actual infinite could be used in a mathematical-metaphysical proof for the existence of God.

This proves that Cantor was a great mathematician, but a poor theologist.

UPDATE: Troy writes in to demonstrate with examples of Cantor's own writings that the attribution above that Cantor denied the existence of "actual infinities" is a blatant lie. Cantor argued with Leibniz that "actual infinities" did physically exist:

"Throughout his tough career at the University of Berlin, Cantor maintained a steady outlook on his adversaries. 'All so-called proofs against the possibility of actually infinite numbers are faulty, as can be demonstrated in every particular case, and as can be concluded on general grounds as well.'" (ref: Georg Cantor, Transfinite Numbers, The Open Court Publishing Company, Chicago and London, 1915.)

There's also an extended quote from the Leibniz Society Review as well that further documents Cantor's position. It's one thing to argue from authority. It's quite another to lie about what the authority said and then argue from that lie.